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          <h2 class="post-title" itemprop="name headline">《机器学习实战》之支持向量机（3）完整版SMO</h2>
        

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<ul>
<li><strong>转载请注明作者和出处：<a href="http://blog.csdn.net/u011475210" target="_blank" rel="noopener">http://blog.csdn.net/u011475210</a></strong></li>
<li><strong>代码地址：<a href="https://github.com/WordZzzz/ML/tree/master/Ch06" target="_blank" rel="noopener">https://github.com/WordZzzz/ML/tree/master/Ch06</a></strong></li>
<li><strong>操作系统：WINDOWS 10</strong></li>
<li><strong>软件版本：python-3.6.2-amd64</strong></li>
<li><strong>编&emsp;&emsp;者：WordZzzz</strong></li>
</ul>
<hr>
<h2 id="前言"><a href="#前言" class="headerlink" title="前言"></a>前言</h2><p>&emsp;&emsp;在小规模数据集上，上一篇文章中的简化版SMO是没有问题的，但是在更大的数据集上，运行速度就会变慢。</p>
<p>&emsp;&emsp;完整版SMO和简化版SMO，实现alpha的更改个代数运算的优化环节一模一样。在优化过程中，唯一的不同就是选择alpha的方式。完整版的SMO算法应用了一些能够提速的启发方法。</p>
<p>&emsp;&emsp;Platt SMO算法通过一个外循环来选择第一个alpha，并且其选择过程会在两种方式之间进行切换：一种是在所有数据集上进行单遍扫描，另一种则是在非边界alpha（不等于边界0或C的alpha值）中实现单遍扫描。对整个数据集的扫描很容易，前面已经实现了，而实现非边界alpha值的扫描时，需要建立这些alpha值得列表，然后再对这个表进行遍历。同时，该步骤会跳过那些已知的不会改变的alpha值。</p>
<p>&emsp;&emsp;在选择第一个alpha值之后，算法会通过一个内循环来选择第二个alpha。在优化过程中，会通过最大化步长的方式来获得第二个alpha值。在简化版SMO算法中，我们会在选择j之后计算错误率Ej。但在这里，我们会建立一个全局的缓存用于保存误差值，并从中选择使得步长或者Ei-Ej最大的alpha值。</p>
<h2 id="支持函数"><a href="#支持函数" class="headerlink" title="支持函数"></a>支持函数</h2><p>&emsp;&emsp;和简化版一样，完整版也需要一些支持函数。</p>
<ul>
<li>首要的事情就是建立一个数据结构来保存所有的重要值，而这个过程可以通过一个对象来完成；</li>
<li>对于给定的alpha值，第一个辅助函数calcEk()能够计算E值并返回（因为调用频繁，所以必须要单独拎出来）；</li>
<li>selectJ()用于选择第二个alpha或者说内循环的alpha值，选择合适的值以保证在每次优化中采用最大步长；</li>
<li>updateEk()用于计算误差值并将其存入缓存中。</li>
</ul>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br></pre></td><td class="code"><pre><span class="line"><span class="string">'''#######********************************</span></span><br><span class="line"><span class="string">Non-Kernel VErsions below</span></span><br><span class="line"><span class="string">'''</span><span class="comment">#######********************************</span></span><br><span class="line"></span><br><span class="line"><span class="class"><span class="keyword">class</span> <span class="title">optStruct</span>:</span></span><br><span class="line">	<span class="string">"""</span></span><br><span class="line"><span class="string">	Function：	存放运算中重要的值</span></span><br><span class="line"><span class="string"></span></span><br><span class="line"><span class="string">	Input：		dataMatIn：数据集</span></span><br><span class="line"><span class="string">				classLabels：类别标签</span></span><br><span class="line"><span class="string">				C：常数C</span></span><br><span class="line"><span class="string">				toler：容错率</span></span><br><span class="line"><span class="string"></span></span><br><span class="line"><span class="string">	Output：	X：数据集</span></span><br><span class="line"><span class="string">				labelMat：类别标签</span></span><br><span class="line"><span class="string">				C：常数C</span></span><br><span class="line"><span class="string">				tol：容错率</span></span><br><span class="line"><span class="string">				m：数据集行数</span></span><br><span class="line"><span class="string">				b：常数项</span></span><br><span class="line"><span class="string">				alphas：alphas矩阵</span></span><br><span class="line"><span class="string">				eCache：误差缓存</span></span><br><span class="line"><span class="string">	"""</span>	</span><br><span class="line">	<span class="function"><span class="keyword">def</span> <span class="title">__init__</span><span class="params">(self, dataMatIn, classLabels, C, toler)</span>:</span></span><br><span class="line">		self.X = dataMatIn</span><br><span class="line">		self.labelMat = classLabels</span><br><span class="line">		self.C = C</span><br><span class="line">		self.tol = toler</span><br><span class="line">		self.m = shape(dataMatIn)[<span class="number">0</span>]</span><br><span class="line">		self.alphas = mat(zeros((self.m, <span class="number">1</span>)))</span><br><span class="line">		self.b = <span class="number">0</span></span><br><span class="line">		self.eCache = mat(zeros((self.m, <span class="number">2</span>)))</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">calcEk</span><span class="params">(oS, k)</span>:</span></span><br><span class="line">	<span class="string">"""</span></span><br><span class="line"><span class="string">	Function：	计算误差值E</span></span><br><span class="line"><span class="string"></span></span><br><span class="line"><span class="string">	Input：		oS：数据结构</span></span><br><span class="line"><span class="string">				k：下标</span></span><br><span class="line"><span class="string"></span></span><br><span class="line"><span class="string">	Output：	Ek：计算的E值</span></span><br><span class="line"><span class="string">	"""</span>	</span><br><span class="line">	<span class="comment">#计算fXk，整个对应输出公式f(x)=w`x + b</span></span><br><span class="line">	fXk = float(multiply(oS.alphas, oS.labelMat).T * (oS.X * oS.X[k,:].T)) + oS.b	</span><br><span class="line">	<span class="comment">#计算E值</span></span><br><span class="line">	Ek = fXk - float(oS.labelMat[k])</span><br><span class="line">	<span class="comment">#返回计算的误差值E</span></span><br><span class="line">	<span class="keyword">return</span> Ek</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">selectJ</span><span class="params">(i, oS, Ei)</span>:</span></span><br><span class="line">	<span class="string">"""</span></span><br><span class="line"><span class="string">	Function：	选择第二个alpha的值</span></span><br><span class="line"><span class="string"></span></span><br><span class="line"><span class="string">	Input：		i：第一个alpha的下标</span></span><br><span class="line"><span class="string">				oS：数据结构</span></span><br><span class="line"><span class="string">				Ei：计算出的第一个alpha的误差值</span></span><br><span class="line"><span class="string"></span></span><br><span class="line"><span class="string">	Output：	j：第二个alpha的下标</span></span><br><span class="line"><span class="string">				Ej：计算出的第二个alpha的误差值</span></span><br><span class="line"><span class="string">	"""</span>	</span><br><span class="line">	<span class="comment">#初始化参数值</span></span><br><span class="line">	maxK = <span class="number">-1</span>; maxDeltaE = <span class="number">0</span>; Ej = <span class="number">0</span></span><br><span class="line">	<span class="comment">#构建误差缓存</span></span><br><span class="line">	oS.eCache[i] = [<span class="number">1</span>, Ei]</span><br><span class="line">	<span class="comment">#构建一个非零列表，返回值是第一个非零E所对应的alpha值，而不是E本身</span></span><br><span class="line">	validEcacheList = nonzero(oS.eCache[:, <span class="number">0</span>].A)[<span class="number">0</span>]</span><br><span class="line">	<span class="comment">#如果列表长度大于1，说明不是第一次循环</span></span><br><span class="line">	<span class="keyword">if</span> (len(validEcacheList)) &gt; <span class="number">1</span>:</span><br><span class="line">		<span class="comment">#遍历列表中所有元素</span></span><br><span class="line">		<span class="keyword">for</span> k <span class="keyword">in</span> validEcacheList:</span><br><span class="line">			<span class="comment">#如果是第一个alpha的下标，就跳出本次循环</span></span><br><span class="line">			<span class="keyword">if</span> k == i: <span class="keyword">continue</span></span><br><span class="line">			<span class="comment">#计算k下标对应的误差值</span></span><br><span class="line">			Ek = calcEk(oS, k)</span><br><span class="line">			<span class="comment">#取两个alpha误差值的差值的绝对值</span></span><br><span class="line">			deltaE = abs(Ei - Ek)</span><br><span class="line">			<span class="comment">#最大值更新</span></span><br><span class="line">			<span class="keyword">if</span> (deltaE &gt; maxDeltaE):</span><br><span class="line">				maxK = k; maxDeltaE = deltaE; Ej = Ek</span><br><span class="line">		<span class="comment">#返回最大差值的下标maxK和误差值Ej</span></span><br><span class="line">		<span class="keyword">return</span> maxK, Ej</span><br><span class="line">	<span class="comment">#如果是第一次循环，则随机选择alpha，然后计算误差</span></span><br><span class="line">	<span class="keyword">else</span>:</span><br><span class="line">		j = selectJrand(i, oS.m)</span><br><span class="line">		Ej = calcEk(oS, j)</span><br><span class="line">	<span class="comment">#返回下标j和其对应的误差Ej</span></span><br><span class="line">	<span class="keyword">return</span> j, Ej</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">updateEk</span><span class="params">(oS, k)</span>:</span></span><br><span class="line">	<span class="string">"""</span></span><br><span class="line"><span class="string">	Function：	更新误差缓存</span></span><br><span class="line"><span class="string"></span></span><br><span class="line"><span class="string">	Input：		oS：数据结构</span></span><br><span class="line"><span class="string">				j：alpha的下标</span></span><br><span class="line"><span class="string"></span></span><br><span class="line"><span class="string">	Output：	无</span></span><br><span class="line"><span class="string">	"""</span>	</span><br><span class="line">	<span class="comment">#计算下表为k的参数的误差</span></span><br><span class="line">	Ek = calcEk(oS, k)</span><br><span class="line">	<span class="comment">#将误差放入缓存</span></span><br><span class="line">	oS.eCache[k] = [<span class="number">1</span>, Ek]</span><br></pre></td></tr></table></figure>
<h2 id="优化例程"><a href="#优化例程" class="headerlink" title="优化例程"></a>优化例程</h2><p>&emsp;&emsp;接下来简单介绍一下用于寻找决策边界的优化例程。</p>
<p>&emsp;&emsp;大部分代码和之前的smoSimple()是一样的，区别在于：</p>
<ul>
<li>使用了自己的数据结构，该结构在oS中传递；</li>
<li>使用selectJ()而不是selectJrand()来选择第二个alpha的值；</li>
<li>在alpha值改变时更新Ecache。</li>
</ul>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">innerL</span><span class="params">(i, oS)</span>:</span></span><br><span class="line">	<span class="string">"""</span></span><br><span class="line"><span class="string">	Function：	完整SMO算法中的优化例程</span></span><br><span class="line"><span class="string"></span></span><br><span class="line"><span class="string">	Input：		oS：数据结构</span></span><br><span class="line"><span class="string">				i：alpha的下标</span></span><br><span class="line"><span class="string"></span></span><br><span class="line"><span class="string">	Output：	无</span></span><br><span class="line"><span class="string">	"""</span>	</span><br><span class="line">	<span class="comment">#计算误差</span></span><br><span class="line">	Ei = calcEk(oS, i)</span><br><span class="line">	<span class="comment">#如果标签与误差相乘之后在容错范围之外，且超过各自对应的常数值，则进行优化</span></span><br><span class="line">	<span class="keyword">if</span> ((oS.labelMat[i]*Ei &lt; -oS.tol) <span class="keyword">and</span> (oS.alphas[i] &lt; oS.C)) <span class="keyword">or</span> ((oS.labelMat[i]*Ei &gt; oS.tol) <span class="keyword">and</span> (oS.alphas[i] &gt; <span class="number">0</span>)):</span><br><span class="line">		<span class="comment">#启发式选择第二个alpha值</span></span><br><span class="line">		j, Ej = selectJ(i, oS, Ei)</span><br><span class="line">		<span class="comment">#利用copy存储刚才的计算值，便于后期比较</span></span><br><span class="line">		alphaIold = oS.alphas[i].copy(); alpahJold = oS.alphas[j].copy();</span><br><span class="line">		<span class="comment">#保证alpha在0和C之间</span></span><br><span class="line">		<span class="keyword">if</span> (oS.labelMat[i] != oS.labelMat[j]):</span><br><span class="line">			L = max(<span class="number">0</span>, oS.alphas[j] - oS. alphas[i])</span><br><span class="line">			H = min(oS.C, oS.C + oS.alphas[j] - oS.alphas[i])</span><br><span class="line">		<span class="keyword">else</span>:</span><br><span class="line">			L = max(<span class="number">0</span>, oS.alphas[j] + oS.alphas[i] - oS.C)</span><br><span class="line">			H = min(oS.C, oS.alphas[j] + oS.alphas[i])</span><br><span class="line">		<span class="comment">#如果界限值相同，则不做处理直接跳出本次循环</span></span><br><span class="line">		<span class="keyword">if</span> L == H: print(<span class="string">"L==H"</span>); <span class="keyword">return</span> <span class="number">0</span></span><br><span class="line">		<span class="comment">#最优修改量，求两个向量的内积（核函数）</span></span><br><span class="line">		eta = <span class="number">2.0</span> * oS.X[i, :]*oS.X[j, :].T - oS.X[i, :]*oS.X[i, :].T - oS.X[j, :]*oS.X[j, :].T</span><br><span class="line">		<span class="comment">#如果最优修改量大于0，则不做处理直接跳出本次循环，这里对真实SMO做了简化处理</span></span><br><span class="line">		<span class="keyword">if</span> eta &gt;= <span class="number">0</span>: print(<span class="string">"eta&gt;=0"</span>); <span class="keyword">return</span> <span class="number">0</span></span><br><span class="line">		<span class="comment">#计算新的alphas[j]的值</span></span><br><span class="line">		oS.alphas[j] -= oS.labelMat[j]*(Ei - Ej)/eta</span><br><span class="line">		<span class="comment">#对新的alphas[j]进行阈值处理</span></span><br><span class="line">		oS.alphas[j] = clipAlpha(oS.alphas[j], H, L)</span><br><span class="line">		<span class="comment">#更新误差缓存</span></span><br><span class="line">		updateEk(oS, j)</span><br><span class="line">		<span class="comment">#如果新旧值差很小，则不做处理跳出本次循环</span></span><br><span class="line">		<span class="keyword">if</span> (abs(oS.alphas[j] - alpahJold) &lt; <span class="number">0.00001</span>): print(<span class="string">"j not moving enough"</span>); <span class="keyword">return</span> <span class="number">0</span></span><br><span class="line">		<span class="comment">#对i进行修改，修改量相同，但是方向相反</span></span><br><span class="line">		oS.alphas[i] += oS.labelMat[j] * oS.labelMat[i] * (alpahJold - oS.alphas[j])</span><br><span class="line">		<span class="comment">#更新误差缓存</span></span><br><span class="line">		updateEk(oS, i)</span><br><span class="line">		<span class="comment">#更新常数项</span></span><br><span class="line">		b1 = oS.b - Ei - oS.labelMat[i] * (oS.alphas[i] - alphaIold) * oS.X[i, :]*oS.X[i, :].T - oS.labelMat[j] * (oS.alphas[j] - alpahJold) * oS.X[i, :]*oS.X[j, :].T</span><br><span class="line">		b2 = oS.b - Ej - oS.labelMat[i] * (oS.alphas[i] - alphaIold) * oS.X[i, :]*oS.X[j, :].T - oS.labelMat[j] * (oS.alphas[j] - alpahJold) * oS.X[j, :]*oS.X[j, :].T</span><br><span class="line">		<span class="comment">#谁在0到C之间，就听谁的，否则就取平均值</span></span><br><span class="line">		<span class="keyword">if</span> (<span class="number">0</span> &lt; oS.alphas[i]) <span class="keyword">and</span> (oS.C &gt; oS.alphas[i]): oS.b = b1</span><br><span class="line">		<span class="keyword">elif</span> (<span class="number">0</span> &lt; oS.alphas[j]) <span class="keyword">and</span> (oS.C &gt; oS.alphas[i]): oS.b = b2</span><br><span class="line">		<span class="keyword">else</span>: oS.b = (b1 + b2) / <span class="number">2.0</span></span><br><span class="line">		<span class="comment">#成功返回1</span></span><br><span class="line">		<span class="keyword">return</span> <span class="number">1</span></span><br><span class="line">	<span class="comment">#失败返回0</span></span><br><span class="line">	<span class="keyword">else</span>: <span class="keyword">return</span> <span class="number">0</span></span><br></pre></td></tr></table></figure>
<h2 id="外循环代码"><a href="#外循环代码" class="headerlink" title="外循环代码"></a>外循环代码</h2><p>&emsp;&emsp;外循环代码的输入和函数smoSimple()完全一样。整个代码的主体是while循环，终止条件：当迭代次数超过指定的最大值，或者遍历整个集合都未对任意alpha对进行修改时，就退出循环。while循环内部与smoSimple()中有所不同，一开始的for循环在数据集上遍历任意可能的alpha。通过innerL()来选择第二个alpha，并在可能时对其进行优化处理。如果有任意一对alpha值发生改变，就会返回1.第二个for循环遍历所有的非边界alpha值，也就是不在边界0或C上的值。</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">smoP</span><span class="params">(dataMatIn, classLabels, C, toler, maxIter)</span>:</span></span><br><span class="line">	<span class="string">"""</span></span><br><span class="line"><span class="string">	Function：	完整SMO算法</span></span><br><span class="line"><span class="string"></span></span><br><span class="line"><span class="string">	Input：		dataMatIn：数据集</span></span><br><span class="line"><span class="string">				classLabels：类别标签</span></span><br><span class="line"><span class="string">				C：常数C</span></span><br><span class="line"><span class="string">				toler：容错率</span></span><br><span class="line"><span class="string">				maxIter：最大的循环次数</span></span><br><span class="line"><span class="string"></span></span><br><span class="line"><span class="string">	Output：	b：常数项</span></span><br><span class="line"><span class="string">				alphas：数据向量</span></span><br><span class="line"><span class="string">	"""</span>	</span><br><span class="line">	<span class="comment">#新建数据结构对象</span></span><br><span class="line">	oS = optStruct(mat(dataMatIn), mat(classLabels).transpose(), C, toler)</span><br><span class="line">	<span class="comment">#初始化迭代次数</span></span><br><span class="line">	iter = <span class="number">0</span></span><br><span class="line">	<span class="comment">#初始化标志位</span></span><br><span class="line">	entireSet = <span class="keyword">True</span>; alphaPairsChanged = <span class="number">0</span></span><br><span class="line">	<span class="comment">#终止条件：迭代次数超限、遍历整个集合都未对alpha进行修改</span></span><br><span class="line">	<span class="keyword">while</span> (iter &lt; maxIter) <span class="keyword">and</span> ((alphaPairsChanged &gt; <span class="number">0</span>) <span class="keyword">or</span> (entireSet)):</span><br><span class="line">		alphaPairsChanged = <span class="number">0</span></span><br><span class="line">		<span class="comment">#根据标志位选择不同的遍历方式</span></span><br><span class="line">		<span class="keyword">if</span> entireSet:</span><br><span class="line">			<span class="comment">#遍历任意可能的alpha值</span></span><br><span class="line">			<span class="keyword">for</span> i <span class="keyword">in</span> range(oS.m):</span><br><span class="line">				<span class="comment">#选择第二个alpha值，并在可能时对其进行优化处理</span></span><br><span class="line">				alphaPairsChanged += innerL(i, oS)</span><br><span class="line">				print(<span class="string">"fullSet, iter: %d i: %d, pairs changed %d"</span> % (iter, i, alphaPairsChanged))</span><br><span class="line">			<span class="comment">#迭代次数累加</span></span><br><span class="line">			iter += <span class="number">1</span></span><br><span class="line">		<span class="keyword">else</span>:</span><br><span class="line">			<span class="comment">#得出所有的非边界alpha值</span></span><br><span class="line">			nonBoundIs = nonzero((oS.alphas.A &gt; <span class="number">0</span>) * (oS.alphas.A &lt; C))[<span class="number">0</span>]</span><br><span class="line">			<span class="comment">#遍历所有的非边界alpha值</span></span><br><span class="line">			<span class="keyword">for</span> i <span class="keyword">in</span> nonBoundIs:</span><br><span class="line">				<span class="comment">#选择第二个alpha值，并在可能时对其进行优化处理</span></span><br><span class="line">				alphaPairsChanged += innerL(i, oS)</span><br><span class="line">				print(<span class="string">"non-bound, iter: %d i: %d, pairs changed %d"</span> % (iter, i, alphaPairsChanged))</span><br><span class="line">			<span class="comment">#迭代次数累加</span></span><br><span class="line">			iter += <span class="number">1</span></span><br><span class="line">		<span class="comment">#在非边界循环和完整遍历之间进行切换</span></span><br><span class="line">		<span class="keyword">if</span> entireSet: entireSet = <span class="keyword">False</span></span><br><span class="line">		<span class="keyword">elif</span> (alphaPairsChanged == <span class="number">0</span>): entireSet =<span class="keyword">True</span></span><br><span class="line">		print(<span class="string">"iteration number: %d"</span> % iter)</span><br><span class="line">	<span class="comment">#返回常数项和数据向量</span></span><br><span class="line">	<span class="keyword">return</span> oS.b, oS.alphas</span><br></pre></td></tr></table></figure>
<p>&emsp;&emsp;测试代码，大家有兴趣的话可以多次运行计算一下运行时间的平均值，看看和简化版相比快了多少。</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">&gt;&gt;&gt; </span>reload(svmMLiA)</span><br><span class="line">&lt;module <span class="string">'svmMLiA'</span> <span class="keyword">from</span> <span class="string">'E:\\机器学习实战\\mycode\\Ch06\\svmMLiA.py'</span>&gt;</span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span>dataArr, labelArr = svmMLiA.loadDataSet(<span class="string">'testSet.txt'</span>)</span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span>b, alphas = svmMLiA.smoP(dataArr, labelArr, <span class="number">0.6</span>, <span class="number">0.001</span>, <span class="number">40</span>)</span><br><span class="line">L==H</span><br><span class="line">fullSet, iter: <span class="number">0</span> i: <span class="number">0</span>, pairs changed <span class="number">0</span></span><br><span class="line">L==H</span><br><span class="line">fullSet, iter: <span class="number">0</span> i: <span class="number">1</span>, pairs changed <span class="number">0</span></span><br><span class="line">fullSet, iter: <span class="number">0</span> i: <span class="number">2</span>, pairs changed <span class="number">1</span></span><br><span class="line">L==H</span><br><span class="line">···</span><br><span class="line">j <span class="keyword">not</span> moving enough</span><br><span class="line">fullSet, iter: <span class="number">2</span> i: <span class="number">97</span>, pairs changed <span class="number">0</span></span><br><span class="line">fullSet, iter: <span class="number">2</span> i: <span class="number">98</span>, pairs changed <span class="number">0</span></span><br><span class="line">fullSet, iter: <span class="number">2</span> i: <span class="number">99</span>, pairs changed <span class="number">0</span></span><br><span class="line">iteration number: <span class="number">3</span></span><br></pre></td></tr></table></figure>
<p>&emsp;&emsp;像之前一样，打印b和alpha，得出的数据用来画图。</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">&gt;&gt;&gt; </span>b</span><br><span class="line">matrix([[<span class="number">-2.89901748</span>]])</span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span>alphas[alphas &gt; <span class="number">0</span>]</span><br><span class="line">matrix([[ <span class="number">0.06961952</span>,  <span class="number">0.0169055</span> ,  <span class="number">0.0169055</span> ,  <span class="number">0.0272699</span> ,  <span class="number">0.04522972</span>,</span><br><span class="line">          <span class="number">0.0272699</span> ,  <span class="number">0.0243898</span> ,  <span class="number">0.06140181</span>,  <span class="number">0.06140181</span>]])</span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span><span class="keyword">from</span> numpy <span class="keyword">import</span> *</span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span>shape(alphas[alphas &gt; <span class="number">0</span>])</span><br><span class="line">(<span class="number">1</span>, <span class="number">9</span>)</span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span><span class="keyword">for</span> i <span class="keyword">in</span> range(<span class="number">100</span>):</span><br><span class="line"><span class="meta">... </span>	<span class="keyword">if</span> alphas[i] &gt; <span class="number">0.0</span>: print(dataArr[i], labelArr[i])</span><br><span class="line"><span class="meta">... </span></span><br><span class="line">[<span class="number">3.542485</span>, <span class="number">1.977398</span>] <span class="number">-1.0</span></span><br><span class="line">[<span class="number">2.114999</span>, <span class="number">-0.004466</span>] <span class="number">-1.0</span></span><br><span class="line">[<span class="number">8.127113</span>, <span class="number">1.274372</span>] <span class="number">1.0</span></span><br><span class="line">[<span class="number">4.658191</span>, <span class="number">3.507396</span>] <span class="number">-1.0</span></span><br><span class="line">[<span class="number">8.197181</span>, <span class="number">1.545132</span>] <span class="number">1.0</span></span><br><span class="line">[<span class="number">7.40786</span>, <span class="number">-0.121961</span>] <span class="number">1.0</span></span><br><span class="line">[<span class="number">6.960661</span>, <span class="number">-0.245353</span>] <span class="number">1.0</span></span><br><span class="line">[<span class="number">6.080573</span>, <span class="number">0.418886</span>] <span class="number">1.0</span></span><br><span class="line">[<span class="number">3.107511</span>, <span class="number">0.758367</span>] <span class="number">-1.0</span></span><br></pre></td></tr></table></figure>
<p>&emsp;&emsp;常数C一方面要保障所有样例的间隔不小于1.0，另一方面又要使得分类间隔要尽可能大，并且要在这两方面之间平衡。如果C很大，那么分类器就会将力图通过分隔超平面对多有的样例都正确分类。这种优化结果如下图，很明显，支持向量变多了。如果数据集非线性可分，就会发现支持向量会在超平面附近聚集成团。</p>
<p></p><br><div align="center"><img src="http://img.blog.csdn.net/20171009201631147?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvdTAxMTQ3NTIxMA==/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast"></div><br><p></p>

<h2 id="分类测试"><a href="#分类测试" class="headerlink" title="分类测试"></a>分类测试</h2><p>好了，终于可以拿我们计算出来的alpha值进行分类了。首先必须基于alpha值得到超平面，这也包括了w的计算。</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">calcWs</span><span class="params">(alphas, dataArr, classLabels)</span>:</span></span><br><span class="line">	<span class="string">"""</span></span><br><span class="line"><span class="string">	Function：	计算W</span></span><br><span class="line"><span class="string"></span></span><br><span class="line"><span class="string">	Input：		alphas：数据向量</span></span><br><span class="line"><span class="string">				dataArr：数据集</span></span><br><span class="line"><span class="string">				classLabels：类别标签</span></span><br><span class="line"><span class="string"></span></span><br><span class="line"><span class="string">	Output：	w：w*x+b中的w</span></span><br><span class="line"><span class="string">	"""</span>	</span><br><span class="line">	<span class="comment">#初始化参数</span></span><br><span class="line">	X = mat(dataArr); labelMat = mat(classLabels).transpose()</span><br><span class="line">	<span class="comment">#获取数据行列值</span></span><br><span class="line">	m,n = shape(X)</span><br><span class="line">	<span class="comment">#初始化w</span></span><br><span class="line">	w = zeros((n,<span class="number">1</span>))</span><br><span class="line">	<span class="comment">#遍历alpha，更新w</span></span><br><span class="line">	<span class="keyword">for</span> i <span class="keyword">in</span> range(m):</span><br><span class="line">		w += multiply(alphas[i]*labelMat[i],X[i,:].T)</span><br><span class="line">	<span class="comment">#返回w值</span></span><br><span class="line">	<span class="keyword">return</span> w</span><br></pre></td></tr></table></figure>
<p>&emsp;&emsp;上述代码中最重要的就是for循环，实现多个数的乘积。虽然for循环遍历了数据集中的所有数据，但是最终起作用的只有支持向量。</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">&gt;&gt;&gt; </span>reload(svmMLiA)</span><br><span class="line">&lt;module <span class="string">'svmMLiA'</span> <span class="keyword">from</span> <span class="string">'E:\\机器学习实战\\mycode\\Ch06\\svmMLiA.py'</span>&gt;</span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span><span class="keyword">from</span> numpy <span class="keyword">import</span> *</span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span>ws = svmMLiA.calcWs(alphas, dataArr, labelArr)</span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span>ws</span><br><span class="line">array([[ <span class="number">0.65307162</span>],</span><br><span class="line">       [<span class="number">-0.17196128</span>]])</span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span>datMat = mat(dataArr)</span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span>datMat[<span class="number">0</span>]* mat(ws)+b</span><br><span class="line">matrix([[<span class="number">-0.92555695</span>]])</span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span>labelArr[<span class="number">0</span>]</span><br><span class="line"><span class="number">-1.0</span></span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span>datMat[<span class="number">1</span>]* mat(ws)+b</span><br><span class="line">matrix([[<span class="number">-1.36706674</span>]])</span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span>labelArr[<span class="number">1</span>]</span><br><span class="line"><span class="number">-1.0</span></span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span>datMat[<span class="number">2</span>]* mat(ws)+b</span><br><span class="line">matrix([[ <span class="number">2.30436336</span>]])</span><br><span class="line"><span class="meta">&gt;&gt;&gt; </span>labelArr[<span class="number">2</span>]</span><br><span class="line"><span class="number">1.0</span></span><br></pre></td></tr></table></figure>
<p>&emsp;&emsp;上面的测试中，计算值大于0属于1类，小于0属于-1类。</p>
<p>&emsp;&emsp;至此，线性分类器介绍完了，如果数据集非线性可分，那么我们就需要引入核函数的概念了，下一篇将进行介绍。</p>
<p><strong><font color="red" size="3" face="仿宋">系列教程持续发布中，欢迎订阅、关注、收藏、评论、点赞哦～～(￣▽￣～)～</font></strong></p>
<p><strong><font color="red" size="3" face="仿宋">完的汪(∪｡∪)｡｡｡zzz</font></strong></p>

      
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